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In this chapter we discuss norm convergence of multiple ergodic averages. We present the beginning of the proof by Host and Kra including the construction of the Host-Kra-Ziegler factors in detail and spend only a few words on the proof of the structure theorem, referring to the works of Host and Kra for details. We present two more proofs of multiple recurrence using properties of the Host-Kra-Ziegler factors, and finally give a brief overview of some further developments and research directions.

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Gowers–Host–Kra Seminorms and Multiple Convergence

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  • 发表日期：

    2025年

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  • Tanja Eisner,
  • Bálint Farkas

摘要

In this chapter we discuss norm convergence of multiple ergodic averages. We present the beginning of the proof by Host and Kra including the construction of the Host-Kra-Ziegler factors in detail and spend only a few words on the proof of the structure theorem, referring to the works of Host and Kra for details. We present two more proofs of multiple recurrence using properties of the Host-Kra-Ziegler factors, and finally give a brief overview of some further developments and research directions.